The logistic map looks simple, but it can produce incredibly complex and chaotic results

The logistic map is one of the classic examples of chaos theory.

"It can be summarised as follows: great complexity may arise from very simple rules," says Olalla Castro Alvaredo of City University London in the UK.

The equation can be used to model many natural processes, for example how a population of animals grows and shrinks over time.

How the population behaves turns out to be enormously sensitive to the value of r, in counterintuitive ways. If r is between 0 and 1 the population will always die, but if it is between 1 and 3 the population will approach a fixed value – and if it is above 3.56995 the population becomes wildly unpredictable.

These behaviours are described as "chaotic" by mathematicians and they are not what we would instinctively expect. But they all emerge from an equation that is mathematically quite simple.

"As we marvel at the diversity and complexity of the natural world, the Universe or the small constituents of matter we should keep remembering that at some fundamental level all of these things share some common simplicity," says Alvaredo.

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